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About the book

Description

Convexity plays an important role in many areas of Mathematics, and this book, the first in a series of three on Convexity and Optimization, studies this concept in detail.

The first half of the book is about convex sets. Convex hull, convex cones, separation by hyperplanes, extreme points, faces, and extreme rays are some of the important notions that are considered. Results for the dual cone are interpreted as solvability criteria for systems of linear inequalities. Closed convex sets in general and polyhedra in particular are characterized in terms of extreme points and extreme rays.

The second half is about convex functions. We study, among other things, convexity preserving operations, maxima and minima of convex functions, continuity and differentiability properties, subdifferentials, and conjugate functions.

The book requires knowledge of Linear Algebra and Calculus of Several Variables.

Content

Preliminaries

Convex sets

Affine sets and affine maps

Convex sets

Convexity preserving operations

Convex hull

Topological properties

Cones

The recession cone

Separation

Separating hyperplanes

The dual cone

Solvability of systems of linear inequalities

More on convex sets

Extreme points and faces

Structure theorems for convex sets

Polyhedra

Extreme points and extreme rays

Polyhedral cones

The internal structure of polyhedra

Polyhedron preserving operations

Separation

Convex functions

Basic definitions

Operations that preserve convexity

Maximum and minimum

Some important inequalities

Solvability of systems of convex inequalities

Continuity

The recessive subspace of convex functions

Closed convex functions

The support function

The Minkowski functional

Smooth convex functions

Convex functions on R

Differentiable convex functions

Strong convexity

Convex functions with Lipschitz continuous derivatives

The subdifferential

The subdifferential

Closed convex functions

The conjugate function

The direction derivative

Subdifferentiation rules

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